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Question Number 30504 by abdo imad last updated on 22/Feb/18
find lim_(x→∞)  x^2 ( e^(1/x)    − e^(1/(x+1)) ) .
findlimxx2(e1xe1x+1).
Answered by sma3l2996 last updated on 24/Feb/18
  lim_(x→∞) x^2 (e^(1/x) −e^(1/(x+1)) )  e^(1/x) ∼_∞ 1+(1/x)+(1/(2x^2 ))   and   e^(1/(x+1)) ∼_∞ 1+(1/(x+1))+(1/(2(x+1)^2 ))  so  e^(1/x) −e^(1/(x+1)) ∼_∞ (1/x)+(1/(2x^2 ))−(1/(x+1))−(1/(2(x^2 +1)))=(1/(x(x+1)))−(1/(2x^2 (x^2 +1)))  lim_(x→∞) x^2 (e^(1/x) +e^(1/(x+1)) )=lim_(x→∞) ((x^2 /(x(x+1)))−(x^2 /(2x^2 (x^2 +1))))  =lim_(x→∞) (x/(x+1))=1
limxx2(e1/xe1/(x+1))e1/x1+1x+12x2ande1/(x+1)1+1x+1+12(x+1)2soe1/xe1/(x+1)1x+12x21x+112(x2+1)=1x(x+1)12x2(x2+1)limxx2(e1/x+e1/(x+1))=limx(x2x(x+1)x22x2(x2+1))=limxxx+1=1

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